Pascal Spincemaille^{1}, Qihao Zhang^{2}, Thanh Dang Nguyen^{3}, and Yi Wang^{3,4}

A vector field perfusion (VFP) approach is proposed to quantify three dimensional blood flow in tissue. The traditional lumped-element Kety equation for characterizing perfusion is fundamentally flawed with the indetermination of arterial input function. The standard continuity equation based on velocity vector field is proposed for fitting tomographic data. Preliminary analysis of dynamic contrast enhanced MRI and arterial spin labelled MRI of the brain demonstrate the feasibility of this VFP approach.

Tissue perfusion is described
using a velocity vector field that satisfies
the mass preservation equation without arterial input function

$$\partial c(\mathbf{r},t)/\partial t=-\nabla\cdot c(\mathbf{r},t)\mathbf{u}(\mathbf{r}) \qquad [1]$$

It is estimated from 4D image data in a manner similar to the optical flow method
in computer vision^{1}. High spatial and temporal resolution CT and MRI
data is fit against Eq $$$[1]$$$ in a Bayesian approach. When image data decay at a known rate $$$\beta$$$, Eq $$$[1]$$$ can be extended
to

$$\partial c(\mathbf{r},t)/\partial t=-\beta c(\mathbf{r},t) -\nabla\cdot c(\mathbf{r},t)\mathbf{u}(\mathbf{r}) \qquad [2]$$

For ASL $$$\beta = 1/T_1$$$. The vector flow $$$\mathbf{f}$$$ into a voxel with cross sectional areas $$$a_x, a_y$$$ and $$$a_z$$$ is then

$$\mathbf{f}(\mathbf{r}) = a_x u_x(\mathbf{r})\mathbf{e}_x + a_y u_y(\mathbf{r})\mathbf{e}_y + a_z u_z(\mathbf{r})\mathbf{e}_z \qquad [3]$$

The traditional Kety flow (ml/100g/min) is obtained as

$$f_K = \|\mathbf{f}\|CBV(100/\rho v)^{2/3} \qquad [4]$$

Here $$$\rho=1.08g/ml$$$ is tissue density, the cross-section scaling factor $$$(100/\rho v)^{2/3}$$$ for 100g tissue, $$$CBV$$$ the cerebral blood volume and $$$v$$$ the voxel volume.

Two types of 4D MRI data were acquired:

Pseudo-continuous ASL (PCASL) 3D FSE data was acquired in 3 healthy subjects using a GE MR750 3T scanner (GEHC, Milwaukee, WI) using a 8 interleave stack of spiral, 512 points/leaf, ±62.5 kHz BW h, 10.5 ms TE, 1.9x1.9x4 mm3 voxel size, 128x128x36 matrix, three signal averages, ~5 min scan time. Five post-label delay times were acquired (1025, 1525, 2025, 1525, 3025 ms). Cerebral blood flow (CBF) maps (ml/100 g/min) were generated using FuncTool (GEHC).

2) MR Perfusion (MRP) Dynamic 3D flow compensated golden ratio stack-of-spiral gradient echo data was acquired in 5 healthy subjects at 1.5T (GEHC) with an 8-channel head coil, 4 echoes, stack-spirals, TR/TEfirst/TElast = 34.3/0.7/25.3 ms, BW = ± 125 kHz, FA = 15˚, matrix size 200×200×22, FOV 22cm, ~2.5 min scan time and 3ml/s injection of 0.1mmol/kg gadobutrol. 3D R2* maps were reconstructed every 748ms using TRACER. PCASL was obtained before contrast.

For
each 4D data set, a cumulative integration across time was done to reduce noise sensitivity and the time intensity curve for
each voxel normalized such that the final frame was equal to 1 for each voxel.
Using
a Bayesian formulation, the resulting 4D data were fit against Eq $$$[2]$$$ with an added gradient based regularization term, which was
done using conjugate gradient..
Arterial CBV was set to
0.74%^{3}.

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